Density Matrix

12 Key excerpts on "Density Matrix"

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  Introduction to Nonlinear Laser Spectroscopy

  • Marc Levenson(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  [1]. describes a system which has probability a% of being found in the ground state |a> and probability b% of being in the excited eigenstate b). All mea-surable properties of a two-level system in this state can be calculated by taking the expectation value of the operator describing the property. For example, the expectation value of the energy is < ψ | Λ -| ψ > = 4£. + # £ » . (2-1.5) The Density Matrix provides an alternative representation of the quantum mechanics of a two-level system. 1 In terms of the wave function Ψ, the Density Matrix for an isolated entity is defined as Ρ = |Ψ><Ψ| (2.1.6) in the representation where a) = ( 0 ) and b} = (?); the Density Matrix equivalent to the general wave function of (2.1.4) is / α Ψ β a l h +iE a t/h L p + i{E b tlK+4>v) 2.1 The Density Matrix for a Two-Level System 31 where Jf is the Hamiltonian of the system. The diagonal elements of ρ represent the probabilities for finding the system in the two basis states (in this representation, the two energy eigenstates). The off-diagonal elements represent the coherence intrinsic to a superposition state. The underlying concepts of coherence and phase are central to nonlinear laser spectroscopy. The real advantage of the Density Matrix is that it can correctly describe the observable properties of an ensemble of quantum-mechanical systems. For an ensemble, the correct definition of the Density Matrix is Ρ = Σ Ρ*|Ψ><Ψ|. (2· 1 · 1 2 ) Ψ where ρ Ψ is the probability of finding the state |Ψ> in the ensemble, and the sum is over all possible quantum states. If every system in the ensemble were known to be in the state |Ψ> of (2.1.4), ρ Ψ = 1 for that state, and the Density Matrix in (2.1.7) would describe the ensemble. Very special preparation is required to achieve such a situation. More generally, the ensemble contains a distribution of quantum states.

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  Introduction to Spintronics

  • Supriyo Bandyopadhyay, Marc Cahay(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  5 The Density Matrix The Density Matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the Density Matrix ). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations . They are intro-duced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T 1 and T 2 , respectively, are also introduced and some numerical examples (results of Monte-Carlo simu-lations) are presented to illustrate their physical meaning. 5.1 Density Matrix concept: Case of a pure state The concept of Density Matrix can be introduced starting with the general ket | ψ angbracketright describing the quantum state of a system associated with a single particle or with an ensemble of particles, as prescribed in the general axioms of quantum-mechanics (see Chapter 18). To make the concept easier to grasp, we restrict ourselves to a general ket in the two-dimensional complex space C 2 , i.e., | ψ angbracketright = α | 0 > + β | 1 >. (5.1) This state is also referred to as a pure state for which the coefficients α and β must satisfy the normalization condition | α | 2 + | β | 2 = 1 . (5.2) Starting with the ket | ψ angbracketright , we define a 2 × 2 Density Matrix as ρ = | ψ angbracketrightangbracketleft ψ | = parenleftbigg α β parenrightbigg ( α ∗ β ∗ ) . (5.3) 91

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  Conceptual Foundations Of Quantum Mechanics

  Second Edition

  • Bernard D'espagnat(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  PART TWO DENSITY MATRICES AND MIXTURES

  In quantum physics, we often need to consider ensembles that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider the ensemble of all the subsystems of some larger systems. The Density Matrix is a very convenient algorithm that makes it possible to deal with such situations swiftly and efficiently.

  This part consists of two chapters. The first one (Chapter 6 ) is an introduction to the mathematical formalism. Its purpose is not to give a systematic review of all the properties of density matrices. Rather, it describes those properties that are most useful for the particular applications we have in mind, that is, the description of mixtures, the various kinds of mixtures, and the problems associated with nonseparability and with the theory of measurement. This is the reason why some known properties of density matrices are not recalled in Chapter 6 and why, on the contrary, some propositions are shown there that are not mentioned in the textbooks.

  The second chapter in this part (Chapter 7 ) provides a description and an examination of the physical concept of mixtures. Along with more conventional aspects of the theory of mixtures, it introduces in particular the notion of proper and of improper mixtures and shows the physical difference between them. It also investigates in what cases an observable can be said to have

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  Principles of Quantum Electronics

  • Dietrich Marcuse(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Once the Density Matrix is known, we can calculate the expectation values of all operators from Eq. (7.1-9) and all probabilities from Eq. (7.2-2). The Density Matrix formalism is thus closely related to the usual quantum mechanics in the Schrödinger picture. In either case we have to solve a differential equation subject to initial conditions. However, the Density Matrix formalism is an extension of quantum mechanics which includes statistical knowledge of incompletely known physical systems. The Density Matrix formalism of quantum mechanics is thus closely analogous to the formalism of classical statistical mechanics [23]. Statistical assumptions are always necessary when the complexity of the problem makes it impossible to obtain all the information about a system, which is possible at least in principle. However, it is not necessary to obtain such detailed information as, for example, the exact velocity of all the molecules of a gas. Average quantities such as the average energy of the molecules which relates to temperature or pressure are all that we are usually interested in. The same situation exists in complex quantum systems, and the Density Matrix formulation is a convenient way to cope with them. The use of this method will become clearer in the following sections. 7 . 3 L a s e r T r e a t e d b y M e a n s o f t h e D e n s i t y M a t r i x Derivation of the Inhomogeneous Wave Equation A simple laser (or maser) model was discussed in Sec. 5.2. In the present section we present the semiclassical theory of a laser (or maser) using the Density Matrix formalism. Our Density Matrix approach of lasers is widely used. However, another popular method of describing lasers (or masers) is based on so-called rate equations that describe how the population of the atomic levels and the photon population vary in either space or time. The rate equation approach, which is usually presented entirely phenomenologically, can be based on the Density Matrix formalism.

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  Quantum Mechanical Irreversibility And Measurement

  • Paolo Grigolini(Author)
  • 1993(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 2 Towards the Statistical Interpretation of Quantum Mechanics This Chapter is devoted to discussing the concept of the quantum mechanical Density Matrix, which turns out to be of central importance to illustrate the so called statistical interpretation of quantum mechanics. Further, we shall show the statistical interpretation at work and discuss both its advantages and limitations. 2.1 The Quantum Mechanical Density Matrix Ballentine in his influential 1970 Reviews of Modern Physics paper (Ballentine, 1970) illustrated in detail the so called Statistical Interpretation of quantum mechanics. According to this survey of the highly dispersed literature on this subject, the related schools of thought can be divided into two major classes. The former group, including John von Neumann and Eugene Wigner, makes the assumption that a state vector provides a complete description of an individual system. Another outstanding scientist belonging to the former group is Werner Heisenberg (Heisenberg, 1958). According to Heisenberg a quantum particle is potentially present' over all regions for which the wavefunction i|/(r) is nonzero, in some intermediate kind of reality, until an act of observation induces a transition from the possible to the actual. The second group, led by Karl Popper (1959, 1969,1985) and including Ballentine himself, rests on the assumption that physical reality is not fully contained in the wavefunction, rather is expressed by ensembles of systems 24 25 prepared in similar ways. An updated survey of the ensemble interpretations 1 of quantum mechanics can be found in the exhaustive review paper by Home and Whitaker (1992). It is convenient to express the Statistical Interpretation of quantum mechanics quoting the precise words used by Ballentine to illustrate his view.

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  Quantum Dynamics

  Applications in Biological and Materials Systems

  • Eric R. Bittner(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  6 Quantum Density Matrix 6.1 INTRODUCTION: MIXED VS. PURE STATES Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state | ψ evolves in time. Using this we can show that the expectation value of an operator evolves as A ( t ) = ψ ( t ) | ˆ A | ψ ( t ) (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ˆ ρ ( t ) = | ψ ( t ) ψ ( t ) | (6.2) taken as the outer product of the state vector with itself. From this definition we can write ˆ ρ = mn c ∗ n c m | m n | (6.3) = mn ρ mn | m n | (6.4) where the ρ mn are the Density Matrix elements. Expectation values of operator are then given by the trace A ( t ) = mn A mn ρ mn = Tr [ ˆ A ρ ( t )] (6.5) where Tr [ ˆ A ρ ( t )] denotes the trace operation: Tr [ ˆ A ρ ( t )] = nn A nn ρ n n If ρ is diagonal, then Tr [ ˆ A ρ ( t )] = n A nn ρ nn = n n | A | n P n where P n = ρ nn is the statistical probability of finding the system in state n . These statistical weights must be such that P n ≤ 1 and n P n = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A. 161 162 Quantum Dynamics: Applications in Biological and Materials Systems The diagonal elements of the Density Matrix, ρ nn = n | ψ ψ | n = p n , are the occupation numbers of the n th state while the off-diagonal ρ nm = n | ψ ψ | m rep-resent the phase coherences between the n and m states. Since the diagonal elements represent probabilities, 1 ≥ ρ nn ≥ 0 so that ρ is a positive semidefinite operator. Moreover, since the trace of a matrix is invariant to representation, the eigenvalues of ρ must be populations as well. Further-more, the eigenvectors of ρ are the pure states of the ensemble. We can define a pure state as a system that can be described by a single-state vector.

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  Emergence of the Quantum from the Classical

  Mathematical Aspects of Quantum Processes

  • Maurice de Gosson(Author)
  • 2017(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  Chapter 6

  Quantum States and the Density Matrix

  In the present chapter, we review the theory of the Density Matrix from a rigorous point of view; this will give us the opportunity to examine the validity of some of the trace formulas we have given in last chapter. This rigor is motivated and needed by the applications to the Density Matrix with variable h we have in mind, and that will be addressed in next chapter. For more on the topics of trace class and Hilbert–Schmidt operators, see for instance [112 , §12.1]; [262 , Appendix 3].

  6.1. The Density Matrix

  The formalism of density operators and matrices was introduced by John von Neumann [226 ] in 1927 and independently, by Lev Landau and Felix Bloch in 1927 and 1946, respectively. Ugo Fano was one of the first to put the theory of the Density Matrix in a rigorous form in his foundational paper [76 ].

  6.1.1. Quantum states

  A quantum system is said to be in a pure state if we have complete knowledge about that system, meaning we know exactly which state it is in. Pure states can be prepared using maximal tests (see [238 , §§2–3]): suppose we are dealing with a quantum system and let N be the maximum number of different outcomes that can be obtained in a test of that system. If such, a test has exactly N different outcomes, it is called a maximal test. The quantum system under consideration is in a pure state if it is prepared in such a way that it certainly yields a predictable outcome in that maximal test; the outcomes in any other test have well-defined probabilities which do not depend on the procedure used for the preparation. A pure state can thus be identified by specifying the complete experiment that characterizes it uniquely (see [76 ]). One usually writes a pure state using Dirac’s ket notation |ψ〉; for all practical purposes it is convenient to use the wavefunction ψ defined by ψ(x) = 〈x|ψ〉; it is a normalized element of a certain Hilbert space , which is usually identified in the case of continuous variables with L2 ( n ) (the square integrable functions). When doing this the state is identified with the linear span of the function ψ, that is the ray ψ = {λψ : λ ∈ }. The pure state |ψ〉 can then be identified with the orthogonal projection ψ of on the subspace ψ. This projection, which is of rank one, is denoted by |ψ〉〈ψ

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  Collapse of the Wave Function

  Models, Ontology, Origin, and Implications

  • Shan Gao(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  It should be emphasized, nevertheless, that, as far as we can see, there is no empirical basis for assuming that individual quantum systems are necessarily in pure states. Nor is there any evidence showing that density matrices necessarily have to be interpreted as representing ensembles. In fact, it seems to be accepted in the quantum infor- mation community that the state of an individual system should be represented, in certain circumstances, by a higher-rank Density Matrix. This can happen, for exam- ple, if the system is entangled with another system and the state of the composite system is pure, in which case the state of the first system is obtained by taking the reduced Density Matrix of the system as a whole, where we trace out the degrees of freedom associated with the second system. It thus seems reasonable to take matters a step further and drop altogether the assumption that individual systems are necessarily in pure states. It also seems reasonable to drop the assumption that 52 Dorje C. Brody and Lane P. Hughston statistical ensembles play a fundamental role in the theory. In our approach, there- fore, we make no use of frequentist thinking, and we avoid reference to observers, measurements, and ensembles. We regard state reduction as an entirely objective phenomenon, and even in the case of an individual system we model the state as a randomly evolving Density Matrix. We denote the Density Matrix process by { ˆ ρ t } t≥0 , and we require that ˆ ρ t should be nonnegative definite for all t and such that tr ˆ ρ t = 1. The dynamical equation generalizing (4.1) then takes the following form: Definition 1 We say that the state { ˆ ρ t } t≥0 of an isolated quantum system with Hamiltonian ˆ H satisfies an energy-driven stochastic master equation with parameter σ if d ˆ ρ t = − i −1 [ ˆ H, ˆ ρ t ]dt + 1 8 σ 2  2 ˆ H ˆ ρ t ˆ H − ˆ H 2 ˆ ρ t − ˆ ρ t ˆ H 2  dt + 1 2 σ  ( ˆ H − H t ) ˆ ρ t + ˆ ρ t ( ˆ H − H t )  dW t , (4.6) where H t = tr ˆ ρ t ˆ H.

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  Interactions of Photons and Neutrons with Matter

  • Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  Chapter 7 THE DENSITY OPERATOR AND ITS ROLE IN QUANTUM STATISTICS Up to this point, we have considered quantum systems where it is assumed that complete knowledge of the state vector exists at some instant of time. Quite often this is not the case; instead one may only have information related to certain macroscopic or thermodynamic properties such as temperature or average energy. Thesesituations can be handled by introducing a so-called density operator into the framework of quantum mechanics. The action of this operator is to assign appropriate probabilities to the occupancy of available quantum states. We illustrate how to determine the density operator of a system by means of the principle of maximum entropy . Aspects of the technique are applied to a few examples such as radiation in a laser cavity both below and above threshold. We conclude this chapter by performing a perturbation expansion of the density operator. Transitions caused by a random perturbation are investigated. 7.1 Mixed States and the Density Operator If a system is in a pure quantum state | ψ angbracketright and one measures a physical observable represented by a Hermitian operator ˆ A , then the probability of obtaining eigenvalue a for the result is P ( a )= |angbracketleft a | ψ angbracketright| 2 . A large number of measurements of the variable A performed on an ensemble of systems, each identically prepared in the state | ψ angbracketright , produces the average result angbracketleft A angbracketright = angbracketleft ψ | ˆ A | ψ angbracketright . However, now consider a situation where one’s a priori knowledge of the system’s quantum state is less than perfect. Suppose, instead, that only the likelihood of being in certain pure states is known. In this case, one says that the system is in a mixed state –a member of the ensemble has a certain probability P ( ψ ) of being in pure state | ψ angbracketright .

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  Statistical Mechanics

  A Set Of Lectures

  • Richard P. Feynman(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 2 DENSITY MATRICES 2.1 INTRODUCTION TO DENSITY MATRICES When we solve a quantum-mechanical problem, what we really do is divide the universe into two parts—the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested com-prised the entire universe. To motivate the use of density matrices, let us see what happens when we include the part of the universe outside the system. Let x describe the coordinates of the system, and let y describe the rest of the universe. Let cpt(x ) be a complete set of wave functions. The most general wave function can be written Hx, y) = £ Ct(y)(plx). (2.1) i At this point we will convert to Dirac notation. Let | (pi) be a complete set of vectors in the vector space describing the system, and let |0t-> be a complete set for the rest of the universe. and et(y) = The most general wave function can be written l^ ) = I Cf> f>|0,.> (2.2) ij ij We can obtain Eq. (2.1) by taking c£ y ) = I c tJo 0 j>. j Now let A be an operator that acts only on the system; that is to say, A does not act on the 1 9j ) . When A acts on product states (for example, |^ » we really mean Aa)b) = (Ad))by. In such a case A does not equal ii' 39 40 Density matrices but equals Z AiAvtyWjXOjKcpii ii’j Then OO = <*w*> = Z Q* c i.r <0j< ij vy = Z ijV = Z <Pi'i (2.3) iV where Pvt = Z c 5c iv = Density Matrix. (2.4) j We define the operator p to be such that p Vi = p operates only on the system described by x. < ^ A ^ y = Z << p A a Z l<i> i V = Z (i> = Tr p A (2.5) i Where we have used the result 2 l^rX^i'l = 1 (by completeness arguments). v From Eq. (2.4), it is obvious that p is hermitian.

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  Statistical Mechanics

  International Series of Monographs in Natural Philosophy

  • R.K. Pathria, D. ter Haar(Authors)
  • 2017(Publication Date)
  • Pergamon

    (Publisher)

  In any other representation, the Density Matrix may or may not be diagonal. However, quite generally, it will be symmetric: Qmn = Qnm^ (13) t It may be noted that in this (so-called energy) representation the density operator ρ may be written as ρ=Σΐ^>ρ»<^»Ι, 02) n for then Qk> = Σ (Ψ* I = 7Ä fc ?J* fc * G ^ A · (14) In terms of the coefficients a%, this becomes ^-iite**'-] .

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  Manipulating Quantum Structures Using Laser Pulses

  • Bruce W. Shore(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  It is important to note that in writing eqn. (16.36) we assume that the matrix elements of ˆ M in the basis ψ n are independent of the environmental parameters e. This assumption is not always valid. For example, if e denotes the velocity of a moving atom and the operator ˆ M depends upon the frequency observed in the rest frame of the atom, then the simplification of eqn. (16.36) is not possible. We then must obtain the expectation value by averaging over density matrices,  ˆ M(t) =   e p(e)  n,m ρ nm (e; t)M mn (e) =   e p(e) Tr[ρ(e; t)M(e)], (16.39) where the elements of the Density Matrix are unaveraged products, ρ nm (e; t) = c n (e; t)c m (e; t) ∗ . (16.40) 3 Because it comprises products of probability amplitudes the Density Matrix contains no information about the overall phase of a statevector. 310 Averages and the statistical matrix (Density Matrix) The statistical operator. The elements of the statistical matrix can be regarded as the matrix elements of a time-dependent statistical operator ˆ ρ(t) in a fixed basis, ρ nm (t) = ψ n | ˆ ρ(t)|ψ m . (16.41) When the statistical operator has the form ˆ ρ(t) = |(t)(t)|, pure state, (16.42) it describes a pure state, whereas when the statistical operator is the weighted sum of such pure-state operators, ˆ ρ(t) =   e |(e; t) p(e) (e; t)|, mixed state, (16.43) it describes a mixed state. From expression (16.43) we evaluate any particular element as ρ nm (t) = ψ n | ˆ ρ(t)|ψ m  =   e p(e)ψ n |(e; t)(e; t)|ψ m , (16.44) in keeping with the definition in eqn. (16.34). 16.6.1 Example: Thermal excitation An important example of a mixed state is the statistical matrix for a system in thermodynamic equilibrium (a thermal state) at temperature T . This statistical operator can be written as ˆ ρ therm = exp(− ˆ H/k B T )/Z (T ), (16.45) where ˆ H is the Hamiltonian energy operator and Z (T ) is the partition function Z (T ) = Tr  exp(− ˆ H/k B T )  .

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